Superconductivity from phonon-mediated retardation in a single-flavor metal

TL;DR

The paper explores whether phonon-mediated pairing can generate unconventional superconductivity in a single-flavor metal when Berry curvature is tunable. It develops a frequency-dependent linearized gap equation in angular-momentum channels, derived from a dynamical phonon-mediated interaction and projected onto a circular Fermi surface. Remarkably, retardation alone yields a leading p-wave instability in the absence of Berry curvature, with a sign-changing gap in frequency and a BCS-like Tc scaling; introducing Berry curvature ($\mathcal{B}>0$) stabilizes chiral p-wave pairing and can drive transitions to higher angular momentum channels, substantially enhancing Tc at optimal curvature. Applying the framework to rhombohedral graphene multilayers, the work links the mechanism to observed quarter-metal superconductivity, estimating Berry-curvature scales $\mathcal{B}k_F^2$ across layers and highlighting the parameter regime where phonon-mediated pairing is most effective.

Abstract

We study phonon-mediated pairings in a single-flavor metal with a tunable Berry curvature. In the absence of Berry curvature, we discover an unexpected possibility: $p$-wave superconductivity emerging purely from the retardation effect, while the static BCS approximation fails to predict its existence. The gap function exhibits sign-change behavior in frequency (owing to the dynamical structure of the phonon-mediated interaction in the $p$-wave channel), and $T_c$ obeys a BCS-like scaling. We further show that the Berry curvature stabilizes the chiral $p$-wave superconductivity and can induce transitions to higher-angular-momentum pairings. Our results establish that the phonon-mediated mechanism is a viable pairing candidate in single-flavor systems, such as the quarter-metal superconductivity observed in rhombohedral graphene multilayers.

Figures (5)

• Figure 1: Angular momentum $L$-channel interactions and gap functions for $\mathcal{B}=0$. (a) The dimensionless interaction $\mathcal{V}_L(\nu_n;k_F,0)/g$ of $L=1,3,5$. The negative value means repulsion due to the convention for pairing interaction. (b)-(d) The rescaled gap function $\bar{\Delta}_L(\omega_n)\equiv\Delta_L(\omega_n)/\Delta_L(\pi T)$ as a function of $\omega_n$. We keep $T_c=10^{-3}\Omega_0$ and use different values of $\lambda$: (b) $\lambda=3.2431$ for $L=1$; (c) $\lambda=11.0439$ for $L=3$; (d) $\lambda=20.3124$ for $L=5$. The black dashed lines indicate zero on the $y$-axis. $4000$ Matsubara frequencies are included in the calculations.
• Figure 2: $T_c$ of $p$-wave SC as a function of the rescaled coupling constant $\tilde{\lambda}$ for $\mathcal{B}=0$. The dimensionless coupling constant $\tilde{\lambda}$ is defined by $\tilde{\lambda}\equiv\lambda/20.7$. Inset: $T_c$ in the logarithmic scale as a function of $1/\tilde{\lambda}$, showing the BCS-like scaling, $T_c\propto\exp(-1/\tilde{\lambda})$.
• Figure 3: $L=1$ interaction potential and gap functions with small $\mathcal{B}k_F^2$. (a) The $L=1$ pairing dimensionless interaction $\mathcal{V}_1/g$ as a function of frequency with several representative values of $\mathcal{B}k_F^2$. (b) The corresponding rescaled gap function $\bar{\Delta}_L(\omega_n)\equiv\Delta_L(\omega_n)/\Delta_L(\pi T)$ as a function of $\omega_n$. We keep $T_c=10^{-3}\Omega_0$ and use different values of $\mathcal{B}k_F^2$. $\lambda=3.2431$ for $\mathcal{B}k_F^2=0$ (blue curve); $\lambda=1.3468$ for $\mathcal{B}k_F^2=0.2$ (cyan curve); $\lambda=0.7591$ for $\mathcal{B}k_F^2=0.4$ (magenta curve); $\lambda=0.5888$ for $\mathcal{B}k_F^2=0.6$ (gold curve).
• Figure 4: $T_c$ as a function of $\mathcal{B}k_F^2$ for $L=1,3,5$. The results show that $T_c$'s are nonmonotonic as tuning $\mathcal{B}k_F^2$. See main text for a detailed discussion. Blue dots: $L=1$; green dots: $L=3$; red dots: $L=5$. The dashed lines indicate the phase transitions between different angular momentum $L$.
• Figure S1: $\mathcal{B}k_F^2$ for rhombohedral graphene multilayers using $k\cdot p$ model. We compute $\mathcal{B}k^2$ (with $k$ relative to the valley momentum) and draw Fermi surface contours (focusing on a single Fermi surface) for the doping density within experimentally superconducting regions. (a) Tetralayer with $V_z=43$ meV and $n_e=0.52\times10^{12}$ cm$^{-2}$. (b) Pentalayer with $V_z=37$ meV and $n_e=0.7\times10^{12}$ cm$^{-2}$. (a) Hexalayer with $V_z=30$ meV and $n_e=0.8\times10^{12}$ cm$^{-2}$. We also average $\mathcal{B}k_F^2$ over the $k$ points near the Fermi surfaces (denoted by $\overline{\mathcal{B}k_F^2}$) for each cases and obtain $\overline{\mathcal{B}k_F^2}\approx 0.47$ for $n=4$ (a), $\overline{\mathcal{B}k_F^2}\approx 1.14$ for $n=5$ (b), and $\overline{\mathcal{B}k_F^2}\approx 1.5$ for $n=6$ (c). Note the different color bar scales in each figure.